Given a compact Lie group $K$ and a maximal torus $T\leq K$, and choose a positive Weyl chamber $\mathfrak t^*_+\subset\mathfrak k^*$, where we used a $K$-invariant inner product on $\mathfrak k$. Then $\mathfrak t^*_+$ can be decomposed into disjoint open faces \begin{equation} \mathfrak t^*_+=\bigsqcup_{\sigma\in\Sigma}\sigma \end{equation} where $\Sigma$ is the set of all faces.

Then question is about centralizer $K_\xi$ of $\xi\in\sigma\in\Sigma$, do we have $K_\xi$ only depends on $\sigma$ and not depends on $\xi\in\sigma$?